Problem: Olivia has taken an initial dose of a prescription medication. The relationship between the elapsed time $t$, in hours, since she took the first dose, and the amount of medication $M(t)$, in milligrams ( $\text{mg}$ ), remaining in her bloodstream is modeled by the following function. $M(t)=50\cdot e^{{-0.75t}}$ How many milligrams of the medication will be remaining in Olivia's bloodstream after $6$ hours? Round your answer, if necessary, to the nearest hundredth.
Thinking about the problem We want to find how many milligrams of the medication will be remaining in Olivia's bloodstream after $6$ hours. In other words, we are given a $t$ value of $6$ and want to find the amount of medication associated with that input, or $M(6)$. To do this, we can substitute $6$ in for $ t$ and evaluate. $M( 6)=50\cdot e^{{-0.75( 6)}}$ Evaluating the expression We can use a calculator to evaluate the expression. The answer is shown below. $\begin{aligned}M(6)&=50\cdot e^{{-0.75(6)}}\\\\ &=50\cdot e^{{-4.5}}\\\\ &\approx0.56\\\\ \end{aligned}$ There are $0.56\text{ mg}$ of medicine remaining in Olivia's bloodstream $6$ hours after the initial dose.